Chapter 6
Part A: Surface analysis – geometrical
methods
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Surface analysis – geometrical methods
 Modelling surfaces - surfaces and fields
Surfaces – typically scalar fields:
 Continuous - z-values (magnitude) assumed to exist for
every (x,y) coordinate pair
 Real valued (may be integer coded, e.g. remote sensing
data) and generally positive (may be negative)
 Single valued (open or 2D manifold) – multiple values
treated as separate surfaces or layers
Surfaces - vector fields:
 Magnitude and direction assumed to exist for every (x,y)
coordinate pair
3rd edition
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Surface analysis – geometrical methods
 Modelling surfaces - surfaces and fields
Mt St Helens – rendered grid
3rd edition
Mt St Helens – wireframe
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Surface analysis – geometrical methods
Modelling surfaces - surfaces and fields
Surfaces - Data sources:
• Physical surfaces – national mapping agencies, field
surveys. DEM, contour, TIN or raster (image) models
plus associated attribute data
• Sample surveys – point/block samples converted to grids
using interpolation procedures
• Remote sensing – satellite, aerial
• Vector data – e.g. wind strength/direction, magnetic
survey data
• Programmatically derived surfaces (theoretical models
and best fits)
3rd edition
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Surface analysis – geometrical methods
Modelling surfaces – raster models
{x,y,z} representation, n x m
Row order – geographic vs mathematical
Treatment of missing and masked data
Coding of cell neighbourhoods
3rd edition
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Surface analysis – geometrical methods
 Modelling surfaces – raster models
Advantages:
 Computationally very convenient
 Easy to display visually (2D image and 3D models)
 Aligns with some data capture (remote sensing) techniques
 Readily available for physical surfaces (DEM)
Disadvantages
 Very large storage requirement
 Computation can be processor intensive
 Fixed grid size, shape, orientation
 Representation of certain objects (e.g. lines) may be poor
3rd edition
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Surface analysis – geometrical methods
Modelling surfaces – raster models
Cell neighbourhoods and derivatives
First order partial derivatives – finite difference model
z z E  zW z zN  zS

,

x
2x
y
2y
z z1,0  z 1,0 z z 0,1  z 0,1

,

x
2 x
y
2 y
Second order partial derivatives (simple version)
z NE  z NW  zSE  zSW
 2 z z E  2 z *  zW  2 z z N  2 z *  zS  2 z

,

,

xy
4 xy
x 2
x 2
y 2
y 2
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Surface analysis – geometrical methods
Modelling surfaces – raster models
Cell neighbourhoods and derivatives
Second order partial derivatives (8-cell finite
difference version)
z z1,1  2z1,0  z1,1   z 1,1  2z 1,0  z 1,1 

x
8x
z z1,1  2 z 0,1  z 1,1   z1,1  2 z 0,1  z 1,1 

y
8y
3rd edition
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Surface analysis – geometrical methods
 Modelling surfaces – raster models
Cell neighbourhoods and derivatives
 Local surface models
• Fit quadratic polynomial to local neighbourhood (OLS)
z=ax2+by2+cxy+dx+ey+f (6 parameters)
• Analytically differentiate
• Aspect: A=tan-1(e/d)
• Slope: St=tan-1(e2+d2)
• Curvatures: see later slides
OR
• Fit partial quartic polynomial to local neighbourhood (exactly)
z=ax2y2+bx2y+cxy2+dx2+ey2+fxy+gx+hy+i (9 parameters)
• Curvatures: see later slides
3rd edition
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Surface analysis – geometrical methods
Modelling surfaces – vector models
Principal models:
TIN
• Compact, fast to process
• Representational detail, complexity of processing
Contour – raster DEM datasets often derived from
contour source material
Conversion to-from TIN/DEM
3rd edition
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Surface analysis – geometrical methods
Modelling surfaces – vector models
A. Source raster
3rd edition
B. Contour - derived
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C. TIN - derived
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Surface analysis – geometrical methods
Modelling surfaces – mathematical models
3rd edition
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Surface analysis – geometrical methods
 Modelling surfaces – statistical and fractal
models
A. Random uniform
3rd edition
B. Random Normal
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C. Ridged multi-fractal
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Surface analysis – geometrical methods
 Modelling surfaces – hybrid (pseudo-random)
models
3rd edition
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Surface analysis – geometrical methods
Surface geometry – gradient, slope,
aspect
Gradient: vector measure – 2 components:
Slope – often computed as rise over run (tan) –
varies by direction. Usually defined as maximum
value at a given point (magnitude component)
Aspect – direction of maximum slope (direction
component)
3rd edition
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Surface analysis – geometrical methods
Surface geometry – slope models
Rise over run (tan)
Rise over surface distance (sin)
Surface z=F(x,y) analytical differential
Surface – grid differential
2
 F 
 F 


S 


 x 
 y 
2
 z  zS
 z  zW 
S  E
   N
 2x 
 2y



2
2
Surface – averaging algorithms (D-infinity, 8-point etc.)
TIN model – direct computation or conversion to grid
Slope – resolution, orientation effects
3rd edition
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Surface analysis – geometrical methods
Surface geometry – aspect
Direction in degrees from North
A  270 
 z z 
360
tan1  , 
2
 x y 
Directional bias from grid orientation
Classified aspect – gradation, 8-way, 4-way
Aspect and lighting/thermal modelling
3rd edition
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Surface analysis – geometrical methods
Surface geometry – profiles
Single profiles
Linear transects
Polygonal transects
3rd edition
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Surface analysis – geometrical methods
Surface geometry – profiles
Multiple profiles
Baselines
are average
across entire
grid
3rd edition
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Surface analysis – geometrical methods
Surface geometry – morphology
3rd edition
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Surface analysis – geometrical methods
 Surface geometry – curvature
 Coordinate systems
1. Original grid coordinates (x,y,z)
2. Rotated grid coordinates (x-rot,y-rot,z) in direction
of aspect
3. Tangential coordinates (surface normal, surface
tangential plane)
 Curvature computation and naming wrt
alternative coordinate systems
3rd edition
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Surface analysis – geometrical methods
Surface geometry – profile curvature
2
Math model:
 pr
 2 z  z 
 2 z z z  2 z
 2
 2
2 
xy x y y
x  x 

pq 3 / 2
2
 z 
 
 y  ,
2
2
 z 
 z 
p       , q  1  p
 x 
 y 
Quadratic model:
Quartic model:
3rd edition
 pr 
 pr 

200 ad 2  be2  cde


e2  d 2 1  d 2  e2


200 dg 2  eh2  fgh
g
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2
 h2


3/2

22
Surface analysis – geometrical methods
Surface geometry – plan curvature
2
Math model:
 pl
 2 z  z 
 2 z z z  2 z  z 
 2
 2

2 
x y x y y  y 
x  x 

p 3/2
2
 z   z 
p   

 x   y 
Quadratic model:
Quartic model:
3rd edition
2
2


e  d 
200  dh2  eg 2  fgh 
 pl 
 g 2  h2 
 pl 
200 bd 2  ae2  cde
2
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23
Surface analysis – geometrical methods
Surface geometry – tangential curvature
2
tg
 2 z  z 
 2 z z z  2 z  z 
 2
 2

2 

x

x

y

x

y
x  
y  y 

pq1/2
2
1/2
p
  pl  
q
2
, where
2
 z   z 
p   
 , and q  1  p
 x   y 
3rd edition
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Surface analysis – geometrical methods
Surface geometry – additional quadratic
curvatures
200  ad  be  cde 
 
Longitudinal:
e  d 
2
lon
2

2
2
200 bd 2  ae2  cde

Cross-sectional:
 cro 
Min, Max and mean:
 min  a  b  (a  b)2  c 2
e
2
 d2

 max  a  b  (a  b)2  c 2
mean  (max  min )/2
3rd edition
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Surface analysis – geometrical methods
Surface geometry – directional derivatives
Computed for direction :
First derivative:
dz z
z

cos() 
sin()
ds x
y
Second derivative:
d 2 z 2 z
2 z
2
 2 cos ( )  2
cos( )sin( )
2
x y
ds
x
2 z 2
 2 sin ( )
y
3rd edition
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Surface analysis – geometrical methods
Surface geometry – paths
Paths as plane curves
Paths as space curves
Parametric specification
Path curvature:
x  2 y y  2 x

2
2

t

t

t

t
 (t) 
3/2
  x 2  y 2 
   

  t   t  


Radius of curvature: 1/path curvature=1/
Smoothing
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Surface analysis – geometrical methods
 Surface smoothing
Resolution increase/Grid
re-calculation
 Using a smoothing
interpolator (e.g. spline)
Filtering or kernel
smoothing (e.g. 3x3
‘Gaussian’ kernel)
3rd edition
1
2
1
2
4
2
1
2
1
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Surface analysis – geometrical methods
Surface geometry – pit filling
Hydrographic modelling
Prior to flow modelling
8-cell model and other rules
Masked fill
Depression-depth based filling
Error correction
Arising from data collection
Arising from data processing (e.g. interpolation)
3rd edition
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Surface analysis – geometrical methods
Surface geometry – volumetric analysis
Profiles – simple cut and fill computations
Surfaces:
Single grid vs reference (base) surface (e.g. z=0)
Grid pairs – grid 1 (upper), grid 2 (lower)
Result – estimate positive or negative volume
(relative, and/or wrt base)
Computational procedures
Numerical integration (trapezoidal rule)
Exact computation from TIN
Indirect computation from point or profile data
3rd edition
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Surface analysis – geometrical methods
Visibility – Overview
Application areas
Line of sight modelling
Viewshed (visible areas) modelling
Single and multi-point problems
Static vs dynamic problems
Optical vs radio path visibility
Euclidean model
Earth curvature model
Propagation modelling
3rd edition
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Surface analysis – geometrical methods
Visibility – line of sight analysis
Simplified form of viewshed
Point source plus direction(s)
Coloured line transect(s)
Tabulated data
Profile plots
Point source, offset
from surface
Viewshed: dark
blue=visible area
Line of sight direction
lines
Lines of sight – yellow=
visible from source,
red=not visible
3rd edition
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Surface analysis – geometrical methods
Visibility – viewsheds and RF propagation
Viewshed (visible areas) modelling
Input surface raster
Point set raster – single, multi-point, zones etc
Offsets for observation and target points
Range (distance and angular) constraints
Output – binary or multi-coded raster
RF – selection of propagation model, parameters
(e.g. frequency, gain) and clutter modelling (typically
surface offsets and obstacles)
3rd edition
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Surface analysis – geometrical methods
 Visibility – viewsheds and RF propagation
A. Source topography
B. Simple optical viewshed
(pink=not visible)
Mobile
phone
mast
3rd edition
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Surface analysis – geometrical methods
Visibility – Isovist analysis
 Analysis of visibility in the plane
 One or more source points
 Complex optimisation problem
Near optimal locations for
cameras providing full
coverage of streets
Sample point – green
areas show visible
street areas
3rd edition
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Surface analysis – geometrical methods
Visibility – Space syntax
 Analysis of visibility in the built environment
3rd edition
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Surface analysis – geometrical methods
Watersheds and drainage – assumptions
Uniform precipitation
Flows take place entirely across surfaces
which they do not alter; unaffected by
absorption or groundwater
Flows grow as a linear function with distance;
not altered by slope values, just by direction
No barriers to flow
Study region is complete and meaningful in
the context of the analysis
3rd edition
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Surface analysis – geometrical methods
Watersheds and drainage – modelling steps
Input (complete/mosaic-ed) DEM
Remove pits
Identify flow directions – D-8, D-infinity or MFM
Output ldd grid
Identify flats and extrema
Accumulate hypothetical flows to generate and
merge streams – include pour points
Identify watersheds and stream basins
3rd edition
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Surface analysis – geometrical methods
Watersheds and drainage – D-infinity
Max gradient of 8 facets identified
Flows assigned to cells (pixels) in proportions:
1
2
p1 
, p2 
1   2
1   2
3rd edition
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Surface analysis – geometrical methods
 Watersheds and drainage – case study
Pit filled DEM
3rd edition
Flow accumulations and watersheds
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Surface analysis – geometrical methods
 Watersheds and drainage – case study
Flats and extrema
3rd edition
Stream basins
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